A Self-Organized Critical Universe

A model of the universe as a self-organized critical system is considered. The universe evolves to a state independently of the initial conditions at the edge of chaos. The critical state is an attractor of the dynamics. Random metric fluctuations exhibit noise without any characteristic length scales, and the power spectrum for the fluctuations has a self-similar fractal behavior. In the early universe, the metric fluctuations smear out the local light cones removing the horizon problem. Typeset using REVTEX 1 One of the most important problems in modern cosmology concerns the fine-tuning necessary in the standard cosmology based on general relativity (GR). Why is the universe so close to being spatially flat after evolving for more than 10 gyr? Why is it so isotropic and homogeneous? How could such a critical state of the universe come about without a severe fine tuning of the parameters? The standard explanation for these questions has been the inflationary models [1]. These models have faced problems that arise mainly from the need to fine tune certain parameters and initial conditions, e.g., the degree of inhomogeneity of the initial universe, or in Linde’s “chaotic” inflation the need to fine tune parameters at the Planck energy. In the following, we shall study a self-organized universe which naturally evolves to a critical state without detailed specification of the initial conditions. The critical state is an attractor of the system which does not need to be fine tuned. This is in contrast to the inflationary models which have an attractor mechanism with critical phase transition points that need to be fine tuned. In statistical mechanics both kinds of attractor mechanism are known to occur for physical systems. In contrast to the inflationary paradigm, we shall be concerned with a self-organized universe in which the spacetime metric fluctuations [2] have a high degree of cooperative effects at the edge of chaos. Recently, the mixmaster universe has been shown to be chaotic, i.e., it does not evolve as a self-organized system in the early universe [3]. In a recent new approach to gravitational theory [2], it has been proposed that at some length scale much larger than the Planck length, lp ≈ 10 −33 cm, the spacetime geometry is fluctuating randomly. In classical GR it is assumed that the spacetime manifold is C smooth down to zero length scales. This seems to be an unacceptably strong hypothesis considering that known physical systems possess dynamical noise at some length scale and that cooperative effects are know to occur for many systems proportional to V 0 and not V −a (a > 0) where V is a characteristic volume for the physical system. The metric tensor was treated as a stochastic variable and for a given three-geometry G, a stochastic differential equation for the momentum conjugate variable was obtained. A Fokker-Planck equation was derived for the probability density leading to statistical mechanical predictions 2 for gravitational systems. Spatially extended dynamical systems with both temporal and spatial degrees of freedom are common in biology, physics and chemistry. These systems can evolve with a spatial structure that develops as a scale-invariant system exhibiting fractal self-similar structure. In statistical mechanics critical phenomena can occur with transition points. Non-equilibrium systems undergo phase transitions with attractors. But in these dynamical systems the critical point can be reached only by a parameter fine-tuning and therefore takes on an accidental description of nature. Inflationary models, as mentioned previously, fall into this catagory of physical systems. But physical systems exist which evolve as self-organized critical structures independent of the initial conditions and with no fine-tuning of the parameters involved in the system. A well-known example is the sand-pile model studied by Bak, Tang and Wiesenfeld [4]. The sand-pile model can also be pictured as a system of coupled damped pendula. Energy is dissipated at all length scales. We shall explore our model of the universe using the Lemâıtre-Tolman-Bondi [5–8] inhomogeneous, spherically symmetric solution of the GR field equations. The metric is given by ds = dt − R(r, t)f(r)dr −R(r, t)dΩ, (1) where f is an arbitrary function of r only, and the field equations demand that R(r, t) satisfies 2RṘ + 2R(1− f ) = F (r) (2) with F being an arbitrary function in classical GR of class C, Ṙ = ∂R/∂t, R = ∂R/∂r, and dΩ = dθ + sin θdφ. The universe is filled with dust and there are three solutions, depending on whether f 2 < 1,= 1, > 1 and they correspond to elliptic (closed), parabolic (flat), and hyperbolic (open) cases, respectively. Let us define the local density parameter, Ω(r, t) = ρ(r, t)/ρc(r, t). There is a correspondence between the spatial curvature and the sign of Ω − 1 : Ω − 1 < 0, f 2 > 1 (open), 3 Ω − 1 = 0, f 2 = 1 (flat), Ω − 1 > 0, f 2 < 1 (closed). The proper distance between two observers is defined by Lr prop = ∫ R(r, t)f(r)dr, (3)